A signal is a potentially varying quantity that can carry information. Some types of signals are waves (e.g., electromagnetic waves, acoustic waves, fluid waves), and accordingly various types of signals include, but are not limited to, electrical signals, optical signals, acoustic signals, pressure signals, and fluid signals. Fluctuations of one or more aspects of a signal generally arise from one or more changes or perturbations in an environment or medium in which the signal is generated and/or propagates. Such signal fluctuations in some instances represent noise and in other instances represent information carried by the signal. Many types of signals may be characterized by their frequency content, i.e., the amount of energy present in the signal at one or more frequencies. This type of characterization is commonly referred to in the relevant arts as a “power spectral density” (PSD) or simply “spectrum” of the signal (e.g., having dimension of Watts/Hz for a voltage signal). The PSD of a signal generally provides valuable insight as to the nature of the information carried by the signal and/or signal noise.
Some types of signals represent “deterministic” processes, in which essentially no randomness is involved in the signal fluctuations. Other types of signals, however, represent random or “stochastic” processes in which the evolution of signal fluctuations over time involves some indeterminacy. Some familiar examples of random processes include: stock market fluctuations; speech, audio or video; and Brownian motion (random movement of particles suspended in a liquid or gas). For a signal representing a random process, the power spectral density of the signal captures the frequency content and thereby helps to identify time correlations or other features in the signal relating to the underlying process. In this manner, the PSD provides a useful way to characterize signal fluctuations representing random processes, including noise and/or information, and in many instances facilitates interpretation of information carried by the signal.
In many practical applications, a given signal is measured or “sampled” via some type of signal detector (e.g., an electrical, optical, or acoustic/pressure detector) over some finite period of time to facilitate signal processing and analysis (e.g., to recover the information carried by the signal). A “sample” of a signal refers to a value or a set of values that is acquired/measured at a particular point in time and/or during a particular interval or window of time. The “sampling frequency” or sampling rate is the number of samples obtained in one second, and the “sampling interval” is the reciprocal of the sampling frequency. The “exposure time” refers to the duration of a given interval or measurement window of time over which one or more values constituting a sample are obtained (i.e., a duration of a measurement window). Thus, for measurement windows that do not overlap in time, the sampling frequency is limited to frequencies at or below the inverse of the exposure time.
The “Nyquist-Shannon” sampling theorem provides a sufficient condition under which reconstruction of a signal from its samples is possible; namely, a band-limited signal having frequency content at some maximum frequency can be reconstructed perfectly if the sampling rate is more than twice the maximum frequency. The frequency equal to one-half the sampling rate is referred to as the “Nyquist frequency” or “Nyquist limit.” From the Nyquist-Shannon theorem, it should be appreciated that the Nyquist limit establishes a conventional bound on the bandwidth of a signal that can be reconstructed/recovered. Accordingly, the Nyquist limit also places constraints on the characterization of signal fluctuations (and the integrity of information) that may be reliably derived from a sampled signal. Further, if the bandwidth of the signal is in excess of the Nyquist limit, a phenomenon commonly referred to as “aliasing” occurs, thereby vitiating the reconstructed/recovered signal.
One of many areas of interest involving random processes and the characterization of signals representing such processes relates to thermal and/or chemical fluctuations associated with microparticles (e.g., molecules, bacteria, colloidal particles, cells). Such microparticles are more generally referred to hereinafter as “objects,” and movements or vibrations of a given object constituting thermal and/or chemical fluctuations are referred to generally as “activity” of the object. The activity of some exemplary microparticles, when optically trapped, has been characterized in the relevant literature as a wide-sense stationary random process, i.e., a random process whose mean and variance do not change over time.
One conventional technique for acquiring signals representing such processes involves particle/object tracking and imaging via video microscopy. According to this technique, an object of interest may be irradiated with radiation (e.g., while confined in an optical trap). In some examples, at least a portion of the radiation is reflected from or transmitted through the object, whereas in other examples the radiation may otherwise interact with the object such that the object itself is an emission source (e.g., the object is stimulated by the incident radiation and as a result itself emits radiation as fluorescence). For purposes of the present disclosure, radiation reflected from, transmitted through, or otherwise emitted by the object all are referred to generally as “radiation from the object.” Radiation from the object optionally may pass through one or more microscope objectives for appropriate magnification, and ultimately impinges on a detector (e.g., an image acquisition device such as a conventional video camera). As the object moves or vibrates while irradiated, the object's motion (i.e., activity) is captured by the video camera. Thus, the radiation from the object and affected by the object's movement constitutes a signal representing a random process (i.e., fluctuations in the object's position), and this signal is sampled by the video camera by virtue of the successive frames of acquired images of the irradiated object.
One challenge presented by this approach is that the video camera acquires images of the object (i.e., samples of the signal) at a sampling rate (e.g., typically around 30 Hz) that is often significantly lower than the rate or rates at which the object may be moving or vibrating; stated differently, standard video camera rates on the order of approximately 30 Hz typically are far lower than the fluctuation frequency of many phenomena of interest. For example, the characteristic timescale for free diffusion of a nanometer/biomolecular-sized object is on the order of nanoseconds (frequencies on the order of gigahertz or GHz). More complex biomolecular processes may be significantly slower, but still faster than standard video rates (e.g., an exemplary bacterial DNA transcription rate is on the order of about 1 kHz).
While faster cameras are available, at increasing expense, it should be appreciated that the intensity of the radiation impinging on the video camera that is required for suitable imaging is inversely proportionally to the exposure time (i.e., duration of a measurement window). Thus, employing a faster video camera with a higher sampling rate and a correspondingly decreased exposure time generally requires significantly higher radiation intensity, which may in turn limit the camera's suitability for many low-light intensity measurement applications (e.g., such as single-molecule fluorescence). The exposure time of a video camera may also affect the camera's frequency response, limiting the frequency up to which fluctuation measurements may be taken. Furthermore, the detectors themselves generally have a frequency response that significantly attenuates at higher frequencies, further reducing the ability of the camera to capture high frequency information (e.g., associated with random processes occurring on a relatively fast timescale). The frequency response of a detector depends on its design and may vary significantly with factors such as wavelength and intensity of the detected radiation. Accordingly, both the sampling frequency/rate of the video camera (which establishes the Nyquist limit) and the frequency response of the camera's detectors place significant constraints on the ability to recover from the sampled signal information that may be associated with frequencies higher than the Nyquist limit and/or the frequency response of the camera's detectors. As such, the power spectral density (PSD) of the sampled signal has an upper frequency bound set by the lower of the Nyquist limit and the detector frequency response.
Recent research efforts relating to thermal and/or chemical fluctuations associated with microparticles have been directed to developing a video microscopy technique to measure the PSD of such a sampled signal above the Nyquist limit in particular circumstances in which a general functional form of the PSD is known a priori. This technique is described in “The effect of integration time on fluctuation measurements: calibrating an optical trap in the presence of motion blur,” by W. Wong and K. Halvorsen, Optics Express, December 2006, Vol. 14, No. 25, 12517, which publication is hereby incorporated herein by reference in its entirety. In this publication, the technique is illustrated using video imaging to characterize the movement/vibrations of a confined particle (e.g., microsphere in an optical trap) at frequencies above the Nyquist limit (which is set by the sampling rate of the video camera used to acquire images of an irradiated particle). Again, this technique requires that the general functional form of the PSD is known a priori, as set forth below.
As noted above, practical acquisition devices collect data (e.g., sample a signal) over a finite exposure time (i.e., duration of a measurement window). In the exemplary video microscopy application discussed above, the radiation from the moving/vibrating object constitutes a signal X(t) representing a wide-sense stationary random process associated with the activity of the object (i.e., the changing position of the object as a function of time). In the simplest case, a sample Xm(t) of the signal is the time average of its true value X(t) over the exposure time W:
                                          X            m                    ⁡                      (            t            )                          =                              1            W                    ⁢                                    ∫                              t                -                                  W                  /                  2                                                            t                +                                  W                  /                  2                                                      ⁢                                          X                ⁡                                  (                                      t                    ′                                    )                                            ⁢                                                ⅆ                                      t                    ′                                                  .                                                                        (        1        )            
Eq. 1 represents a phenomenon commonly referred to in the relevant arts as the “integration effect” or “aperture effect.” Here the timestamp, t, is defined as the average time in the measurement window. In the example of video imaging, averaging leads to the common problem of video image blur, in which object motion during the measurement window often causes errors in the signal sample Xm (i.e., the sampled position of the object). Image blur increases the measurement uncertainty and adds systematic biases to the measured values, thereby distorting the frequency content of the signal being sampled. Generally, as noted above, frequency information associated with the sampled signal is reliable only for frequencies below the lower of the Nyquist limit and the maximum frequency response of the sampling device (e.g., the radiation detectors of the video camera), and for sufficiently short exposure times W that render the aperture effect essentially negligible.
The PSD of the actual signal X(t) (representing a real-valued trajectory for a wide-sense stationary random process) is defined as the Fourier transform of the auto-correlation function. For a wide-sense stationary random process the auto-correlation function of X is solely a function of the time shift, τ:RXX(τ)=X(t)X(t+τ)  (2)where the angle brackets  . . .  represent an ensemble average. The PSD is therefore given by:P(ω)≡{tilde over (R)}XX(ω).  (3)The tilde is used to designate the Fourier transform, {tilde over (X)}(ω)=∫X(t)exp(−iωt)dt. The integral is taken over the entire integration domain (e.g., from −∞ to +∞) unless otherwise specified.
Due to the integration or aperture effect, which leads to image blur as noted above, the PSD derived from the sampled signal Pm(ω) differs from the PSD P(ω) of the actual signal. The difference may be quantified by first expressing the signal sample Xm(t) as a convolution with an “impulse response” in the form of a rectangular function representing the measurement window:Xm(t)=(X*H)(t)≡∫X(t′)H(t−t′)dt′,  (4)where H(t) is the impulse response (i.e., rectangular function):
                              H          ⁡                      (            t            )                          =                  {                                                                      1                  W                                                                                                                        -                      W                                        /                    2                                    <                  t                  ≤                                      W                    /                    2                                                                                                      0                                                              elsewhere                  .                                                                                        (        5        )            This is equivalent to the time integral in Eq. 1. Eq. 2 and Eq. 4 may then be employed to determine the autocorrelation function of Xm, namely RXmXm, from which the PSD of the sampled signal, Pm(ω), is given by:Pm(ω)≡{tilde over (R)}XmXm(ω).  (6)Equivalently, the PSD of the sampled signal may be expressed in terms of the PSD of the actual signal P(ω) and the Fourier transform of the rectangular impulse response representing the measurement window, according to:Pm(ω)=P(ω)|{tilde over (H)}(ω)|2.  (7)
While the direct measurement of Pm(ω) may be made up to the Nyquist frequency, the variance effectively integrates over all frequencies:
                                          var            ⁡                          [                              X                m                            ]                                ⁢                      (            W            )                          =                              1                          2              ⁢              π                                ⁢                      ∫                                                            P                  m                                ⁡                                  (                  ω                  )                                            ⁢                              ⅆ                ω                            ⁢                                                          ⁢                              (                8                )                                                                            =                              1                          2              ⁢              π                                ⁢                      ∫                                                            P                                      R                    i                                                  ⁡                                  (                  ω                  )                                            ⁢                                                                                                                                    H                        ~                                            W                                        ⁡                                          (                      ω                      )                                                                                        2                            ⁢                                                ⅆ                  ω                                .                                                                  ⁢                                  (                  9                  )                                                                        Superscripts have been added in Eq. 9 to explicitly indicate that {tilde over (H)} is a function of the exposure time W, and P is a function of some physical parameters, Ri. The variance is written as a function of the exposure time, W, to further emphasize that if the form of PRi(ω) as a function of Ri is known, then these parameters may be determined by measuring the variance at different values of W while keeping all the physical settings fixed, then performing a non-linear fit to var[Xm](W) with Ri as the fitting parameters.
As can be seen from Eq. 10 below, the detection bandwidth using this method is limited not by the sampling rate of the acquisition system, but rather by the maximum shutter speed 1/W (i.e., minimum exposure time) of the video camera.
                                                                                                      H                  ~                                W                            ⁡                              (                ω                )                                                          2                =                              (                                          sin                ⁡                                  (                                      ω                    ⁢                                                                                  ⁢                                          W                      /                      2                                                        )                                                            ω                ⁢                                                                  ⁢                                  W                  /                  2                                                      )                    2                                    (        10        )            
In summary, determination of the power spectral density, PRi(ω), requires solving for the physical parameters defining the assumed functional form of PRi(ω). The functional form is not determined from this measurement, but may be known, for example, from a theoretical model for the process being measured. Solving for the physical parameters using the experimentally measured variances, var[Xm](W), firmly defines the scale and shape of the power spectral density of the process for the specific experimental conditions. The accuracy of the power spectral density is dependent on an appropriate choice of the functional form of PRi(ω) and the stability of the process during measurements such that the defining physical parameters are constant.